Two versions of pseudo-differential operators involving the Kontorovich Lebedev transform in L 2 (R + ; dx

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1 Forum Math. 218; 3(1): Research Article Akhilesh Prasad* and Uain K. Mandal Two versions of seudo-differential oerators involving the Kontorovich Lebedev transform in L 2 (R + ; d ) DOI: /forum Received December 15, 216 Abstract: The Pseudo-differential oerators (.d.o.) L(, A ) and L(, A ) involving the Kontorovich Lebedev transform are defined. An estimate for these oerators in the Hilbert sace L 2 (R + ; d ) is obtained. A symbol class Λ is defined and it is shown that the roduct of any two symbols from this class is again in Λ. At the end, commutators for the.d.o. and their boundedness results are discussed. Keywords: Pseudo-differential oerator, Kontorovich Lebedev transform, Hilbert sace MSC 21: 35S5, 44A2 Communicated by: Christoher D. Sogge 1 Introduction In 1938, M. I. Kontorovich and N. N. Lebedev introduced the Kontorovich Lebedev transform (KL-transform) that has been used to solve boundary value roblems like in diffraction theory and electronamics [8, 9]. Further, it has been used by many researchers to solve hysical roblems; see for instance [11, 22]. Moreover, KL-transforms also have been studied in the field of ure mathematics [1, 12, 19, 25, 27, 29, 3]. The KL-transform of a function φ() defined on the ositive half line R + = (, ), as given in [1, 23] is defined as (Kφ)(τ) = K iτ ()φ() 1 d, τ R +, (1.1) where K iτ () reresents the Macdonald function (modified Bessel s function of second kind) and it is given (see [2,. 82 (21)]) as K iτ () = e cosh t cos(τt) dt, >, τ >. Also, K iτ () satisfies the differential euation A K iτ () = τ 2 K iτ (), where the differential oerator A is defined (see [3, 4, 13]) as A = 2 d2 d 2 + d d 2. Its adjoint oerator is given as A = d2 d 2 2 d d 2. *Corresonding author: Akhilesh Prasad: Deartment of Alied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad-8264, India, ar_bhu@yahoo.com Uain K. Mandal: Deartment of Alied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad-8264, India, uainmandal@gmail.com

2 32 A. Prasad and U. K. Mandal, Pseudo-differential oerators involving KL-transform Also, A ( 1 φ()) = 1 A φ(). Moreover, the reresentation of A n φ() can be written (see [3, 4, 13]) as A n φ() = 2n j= j P n j ()Dj φ() for all n N, where the P n j are olynomials of degree 2n j for even j and 2n j 1 for odd j resectively. Thus P2n n () = 1 and P2n 1 n () = n(2n 1). The inversion of (1.1) is given by φ() = K 1 (Kφ)() = 2 π 2 K iτ ()(Kφ)(τ)τ sinh(πτ) dτ, R +. (1.2) From [1, 23, 29], for K : L 2 (R + ; d ) L2 (R + ; τ sinh(πτ)) the Plancherel relation is given as φ() ψ() d = 2 π 2 (Kφ)(τ)(Kψ)(τ)τ sinh(πτ) dτ. (1.3) The Parseval formula for the KL-transformation (1.1), is given as φ() 2 d = 2 π 2 (Kφ)(τ) 2 τ sinh(πτ) dτ. (1.4) Other tyes of Plancherel relations and Parseval relations for the Kontorovich Lebedev transform can be found in [1, 12, 19, 26, 29]. From [2], we have 2 π 2 K iτ ()K iτ (y)k iτ ()τ sinh(πτ) dτ = T(, y, ), where T(, y, ) is symmetric in, y, and defined as From [17,. 344], we have Thus for α =, = 1 2 T(, y, ) = 1 2 e[ 1 2y (2 y 2 + y )],, y, R +. α 1 e α d = 2( ) 2 K α (2 ), Re(), Re() >, α R. 2 +y 2 y and = 1 2 y, in the above euation we get T(, y, ) 1 d = K ( 2 + y 2 ). (1.5) Also from [25, 26], we have K ( 2 + y 2 ) K () or K (y). Further from [18,. 354], we have 1 e Kμ (c) d = 2K μ (A + )K μ (A ), where A ± = ( + c ± c), Re( ± c) and Re() >. Thus for μ =, = 1 2 the above euation, we get 2 +y 2 y, c = 1, and = 1 2 y in T(, y, )K () 1 d = K ()K (y). (1.6)

3 A. Prasad and U. K. Mandal, Pseudo-differential oerators involving KL-transform 33 The convolution oerator is now defined by (φ ψ)() = T φ(y)ψ(y)y 1 where the translation oerator T φ(y) is given as = T(, y, )φ()ψ(y) 1 y 1 d, (1.7) T φ(y) = T(, y, )φ() 1 d. This aer contains five sections. Section 1 is introductory. In Section 2, estimates of convolution related to the KL-transform are obtained. In Sections 3 and 4, two seudo-differential oerators (.d.o.) L(, A ) and L(, A ) are introduced and estimates of these oerators in L 2 (R + ; d ) are obtained. Moreover, the oerators L(, A ) and L(, A ) are adjoint to each other as is shown in Section 4. Lastly, Section 5 deals with the roduct and commutators of these.d.o., and their boundedness results are discussed. 2 Estimates of convolution Kontorovich Lebedev transforms have been defined in various ways [14, 25, 26] and estimates of convolution associated with them were discussed reviously by many authors; for instance see [5, 25, 26, 28]. In this section, we obtain some new ineualities for convolution (1.7). Theorem 2.1. If φ L (R + ; K () d ), 1 < < and ψ L1 (R + ; K () d ), then φ ψ L (R + ;K () d ) φ L (R + ;K () d ) ψ L 1 (R + ;K () d ). Proof. Now from (1.7) and using Hölder s ineuality, we get (φ ψ)() ( (φ ψ)() ( Therefore, using Fubini s theorem, we have T(, y, ) φ() ψ(y) d T(, y, ) φ() ψ(y) d (φ ψ)() K () d ( ( Now using (1.5) and (1.6), we get (φ ψ)() K () d ( ( y ) 1 ( y )( T(, y, ) ψ(y) d T(, y, ) ψ(y) d y ) 1, y ). T(, y, )K () d ) φ() ψ(y) d y ) K (y)k () φ() ψ(y) d K (y)k () φ() ψ(y) d T(, y, ) d ) ψ(y) y ). y ( y ( ( φ() K () d )( ψ(y) K (y) y ). K ( 2 + y 2 ) ψ(y) y ) K (y) ψ(y) y )

4 34 A. Prasad and U. K. Mandal, Pseudo-differential oerators involving KL-transform Therefore, (φ ψ) L (R + ;K () d ) φ L (R + ;K () d ) ψ L 1 (R + ;K () d ). Theorem 2.2. If φ L (R + ; d ), 1 < <, and ψ L (R + ; d ), 1 < <, with = 1, then where φ ψ L 1 (R + ;d) π 2 φ L (R + ; d ) ψ L (R + ; ), (2.1) φ ψ L r (R + ; r 1 d) C r φ L (R + ; d ) ψ L (R + ; ), (2.2) C r = [ 1 K r r ()r 1 d], 1 r <. Proof. Proof of 2.1: Using (1.7) and Hölder s ineuality, we get (φ ψ)() ( T(, y, ) φ() d Now using Fubini s theorem and (1.5), we have Thus (φ ψ)() ( y ) 1 K ( ) φ() d ) 1 K ()( φ() d 1 ) Proof of 2.2: From (2.3), the roof is straightforward. ( ( ( y y T(, y, ) ψ(y) d y ) 1. K ( 2 + y 2 ) ψ(y) 1 y ) ψ(y) 1 y ). (2.3) φ ψ L 1 (R + ;d) π 2 φ L (R + ; d ) ψ L (R + ; y ). 3 The seudo-differential oerator L(, A ) In the mid-6s, seudo-differential oerators were studied by Kohn and Nirenberg [7] and Hörmander [6] by using the theory of Fourier transforms and then seeing its imortance in the theory of artial differential euations [2, 24, 31]. Some seudo-differential oerators associated with other integral transformations like Hankel transformations, Fourier Jacobi transformations etc. are defined and their roerties are discussed in [13, 15, 16, 21]. Motivated by the works of Zaidman [31] and Pathak and Uadhyay [15], we define the seudo-differential oerator L(, A ) involving the KL-transform for any function φ L 2 (R + ; d ) as where with (L(, A )φ)() = 2 π 2 K iτ ()G(τ)τ sinh(πτ) dτ, (3.1) G(τ) = l(, τ)(kφ)(τ) + l τ(η)(kφ)(η)η sinh(πη) dη (3.2) l τ(η) = 2 π 2 K iτ ()K iη ()l (, τ) d. (3.3) Here, the function l(, τ) C (R + R + ) is said to be an element of the class Λ if and only if l(, tτ) = l(, τ) for t >, lim l(, τ) = l(, τ) eists for τ R + and l(, τ) is a C function. We define l (, τ) = l(, τ) l(, τ) (3.4)

5 A. Prasad and U. K. Mandal, Pseudo-differential oerators involving KL-transform 35 and assume the estimate as (1 + ) n D α D β τl (, τ) C α,β,n for all R +, (3.5) where τ = 1 and α, β, n N. For the eistence of (3.1) we shall show that K 1 G eists. Using the definition of Λ, we have and Thus using (1.4), we can write Hence l(, τ)(kφ)(τ) l(, 1) (Kφ)(τ) l(, τ)(kφ)(τ) L 2 (R + ;τ sinh(πτ)) l(, 1) (Kφ)(τ) L 2 (R + ;τ sinh(πτ)). (K 1 (l(, τ)(kφ)(τ)))() L 2 (R + ; d ) = l(, τ)(kφ)(τ) L 2 (R + ;τ sinh(πτ) dτ). (K 1 (l(, τ)(kφ)(τ)))() L 2 (R + ; d ). Now for the eistence of K 1 G it is sufficient to show that (K 1 ( Using (3.3), (1.2) and Fubini s theorem, we get thus, l τ(η)(kφ)(η)η sinh(η) dη))() L 2 (R + ; d ). l τ(η)(kφ)(η)η sinh(πη) dη = K iτ ()l (, τ)( 2 π 2 = K iη ()(Kφ)(η)η sinh(πη) dη) d K iτ ()l (, τ)φ() d = (K(φ()l (, τ)))(τ), (3.6) (K 1 ( l τ(η)(kφ)(η)η sinh(η) dη))() = φ()l (, τ). Now as φ L 2 (R + ; d ) and l (, τ) satisfies (3.5), we can write φ()l (, τ) C n (1 + ) n φ(), for n N and R + such that (1 + ) n is a ositive constant. Thus we have where C n is some ositive constant. Thus (K 1 ( φ()l (, τ) L 2 (R + ; d ) C n φ() L 2 (R + ; d ), (3.7) l τ(η)(kφ)(η)η sinh(πη) dη))() L 2 (R + ; d ) and hence we conclude that (K 1 G)() = (L(, A )φ)() eists as an element of L 2 (R + ; d ). From (3.1), we can also write (K(L(, A )φ()))(τ) = l(, )(Kφ)(τ) + l τ(η)(kφ)(η)η sinh(πη) dη. Another reresentation of L(, A ) is given as follows.

6 36 A. Prasad and U. K. Mandal, Pseudo-differential oerators involving KL-transform Theorem 3.1. If l(, τ) Λ, then for φ L 2 (R + ; d ) we have Proof. By definition of Λ, we have Now, Moreover, from (3.6) we have (L(, A )φ)() = 2 π 2 K iτ ()( Adding (3.1), (3.11) and using (3.9), we get K iτ (y)l(y, τ)φ(y) )τ sinh(πτ) dτ. (3.8) y l(, τ) = l(, τ) + l (, τ). (3.9) K iτ (y)l(, τ)φ(y) y = l(, τ)(kφ)(τ). (3.1) K iτ (y)l (y, τ)φ(y) y = l τ(η)(kφ)(η)η sinh(πη) dη. (3.11) K iτ (y)l(y, τ)φ(y) y = G(τ) = (KL(, A )φ)(τ). (3.12) Also, as earlier, K 1 G = L(, A )φ L 2 (R + ; d ). Therefore using (1.4), we have K 1 G L 2 (R + ; d ) = G L 2 (R + ;τ sinh(πτ) dτ). Thus, G(τ) eists as an element of L 2 (R + ; τ sinh(πτ) dτ). Hence, in view of (1.2), from (3.12) we get the reuired result (3.8). Theorem 3.2. Let l(, τ) Λ and L(, τ)φ be defined as in (3.4). Then where E n is a ositive constant. (L(, A )φ)() L 2 (R + ; d ) E n φ() L 2 (R + ; d ), (3.13) Proof. We have Thus using (3.1) and (3.2), we have L(, A ) = L(, A ) + L (, A ). Thus using (1.4), we have and Now from (3.6), we have (K(L(, A )φ))(τ) = l(, τ)(kφ)(τ). L(, A )φ L 2 (R + ; d ) l(, 1) φ L 2 (R + ; d ) (3.14) (K(L (, A )φ))(τ) = l τ(η)(kφ)(η)η sinh(η) dη. L (, A )φ)() = φ()l (, τ). Using (3.7), we get (L (, A )φ)() L 2 (R + ; d ) C n φ L 2 (R + ; d ), (3.15) where C n > is a constant. Hence using (3.14) and (3.15), we get (L(, A )φ)() L 2 (R + ; d ) = (L(, A )φ)() + (L (, A )φ)() L 2 (R + ; d ) where E n = ma( l(, 1), C n) is a ositive constant. l(, 1) φ L 2 (R + ; d ) + C n φ L 2 (R + ; d ) E n φ() L 2 (R + ; d ),

7 A. Prasad and U. K. Mandal, Pseudo-differential oerators involving KL-transform 37 4 The seudo-differential oerator L(, A ) In this section, we define the.d.o. L(, A ) associated with the symbol l(, D) Λ. Later on, we shall show that L(, A ) and L(, A ) are adjoint to each other. For any function φ in L 2 (R + ; d ), the oerator L(, A ) is defined as where G(τ) is given as with (L(, A )φ)() = 2 π 2 K iτ ()G(τ)τ sinh(πτ) dτ, (4.1) G(τ) = l(, τ)(kφ)(τ) + l η(τ)(kφ)(η)η sinh(πη) dη (4.2) l η(τ) = 2 π 2 K iτ ()K iη ()l (, η) d. (4.3) Now following the same argument as done reviously for L(, A )φ, we can rove the eistence of L(, A )φ in L 2 (R + ; d ) for any function φ L2 (R + ; d ). An alternative reresentation of L(, A ) is given in the following theorem. Theorem 4.1. For any function φ L 2 (R + ; d ) and l(, τ) Λ we have Proof. From (4.3), we can write Now, Thus from (4.1), (4.2) and (4.5), we get (L(, A )φ)() = 2 π 2 K iτ ()l(, τ)(kφ)(τ)τ sinh(πτ) dτ. (4.4) (K 1 l η(τ))() = 2 π 2 K iη()l (, η). 2 π 2 K iτ ()( l η(τ)(kφ)(η)η sinh(πη) dη)τ sinh(πτ) dτ = ( 2 π 2 K iτ ()l η(τ)τ sinh(πτ) dτ)(kφ)(η)η sinh(πη) dη = 2 π 2 K iη ()l (, τ)(kφ)(η)η sinh(πη) dη. (4.5) (L(, A )φ)() = 2 π 2 K iτ ()[l(, τ)(kφ)(τ) + l η(τ)(kφ)(η)η sinh(πη) dη]τ sinh(πτ) dτ = 2 π 2 K iτ ()l(, τ)(kφ)(τ)τ sinh(πτ) dτ + 2 π 2 K iη ()l (, η)(kφ)(η)η sinh(πη) dη. By a change of variable in the second integral and using the definition of the symbol class Λ, we have (L(, A )φ)() = 2 π 2 K iτ ()l(, τ)(kφ)(τ)τ sinh(πτ) dτ.

8 38 A. Prasad and U. K. Mandal, Pseudo-differential oerators involving KL-transform Theorem 4.2. Let l(, τ) Λ and L(, A )φ be defined as in (4.1). Then for any φ L 2 (R + ; d ). Proof. The roof is similar to the one of Theorem 3.2. L(, A )φ L 2 (R + ; d ) E n φ L 2 (R + ; d ) (4.6) Theorem 4.3. For L(, A ) and L(, A ) defined as in (3.1) and (4.1), resectively, we have where E n is a ositive constant. Proof. Using (3.13) and (4.6), we get (L(, A ) L(, A ))φ L 2 (R + ; d ) E n φ L 2 (R + ; d ), (L(, A ) L(, A ))φ L 2 (R + ; d ) L(, A )φ L 2 (R + ; d ) + L(, A )φ L 2 (R + ; d ) E n φ L 2 (R + ; d ) + E n φ L 2 (R + ; d ) where E n = 2E n. E n φ L 2 (R + ; d ), Theorem 4.4. Let l(, τ) be a symbol and let l(, τ) be its comle conjugate. Suose L(, A ) and L(, A ) are the oerators associated with the symbols l(, τ) and l(, τ), resectively. Then where φ, ψ L 2 (R + ; d ). Proof. Using (1.3), we get L(, A )φ, ψ L 2 (R + ; d ) = φ, L(, A )ψ L 2 (R + ; d ), Now, using (3.8), we have Also, using (4.4), we have Thus, which is similar to (4.7). L(, A )φ, ψ L 2 (R + ; d ) = 2 π 2 K(L(, A )φ)(τ), (Kψ)(τ) L 2 (R + ;τ sinh(πτ) dτ). L(, A )φ, ψ L 2 (R + ; d ) = 2 π 2 K iτ (y)l(y, τ)φ(y)(kψ)(τ)τ sinh(πτ) dτ y. (4.7) (L(, A )ψ)() = 2 π 2 K iτ ()l(, τ)(kψ)(τ)τ sinh(πτ) dτ. φ, L(, A )ψ L 2 (R + ; d ) = 2 π 2 K iτ ()l(, τ)(kψ)()φ()τ sinh(πτ) dτ d, 5 Product and commutators Theorem 5.1. If l(, τ), b(, τ) Λ, then c(, τ) = l(, τ)b(, τ) Λ. Proof. Since l(, τ), b(, τ) C (R + R + ), we have c(, τ) C (R + R + ). By the definition of the symbol class Λ, for all t > we have c(, tτ) = l(, tτ)b(, tτ) = l(, τ)b(, τ) = c(, τ) and that lim l(, τ) = l(, τ), lim τ b(, τ) = b(, τ),

9 A. Prasad and U. K. Mandal, Pseudo-differential oerators involving KL-transform 39 eist for τ R +. Also, eists for τ R +. Thus, if we ut we have where lim c(, τ) = c(, τ) = l(, τ)b(, τ) c (, τ) = c(, τ) c(, τ), c(, τ) = (l (, τ) + l(, τ))(b (, τ) + b(, τ)) = l (, τ)b (, τ) + l(, τ)b (, τ) + l (, τ)b(, τ) + l(, τ)b(, τ) = c (, τ) + c(, τ), c (, τ) = l (, τ)b (, τ) + l(, τ)b (, τ) + l (, τ)b(, τ). Net, we show that c (, τ) satisfies (3.5). Using Leibni s theorem, we have (1 + ) n D α D β τc (, τ) (1 + ) n D α D β τl (, τ)b (, τ) + (1 + ) n D α D β τl(, τ)b (, τ) + (1 + ) n D α D β τb(, τ)l (, τ) α s= α s= ( α β s ) + + M α,β,n. r= α s= α s= ( β r )(1 + )n D α s ( α β s ) r= ( α β s ) r= ( β r )(1 + )n D α s ( β r )(1 + )n D α s D β r τ l (, τ) D s D r τb (, τ) ( α β s ) ( β r )[C 1,α,β,n + C 2,α,β,n + C 3,α,β,n ] r= Hence from (3.4) and (3.5), we have c(, τ) Λ. D β r τ b (, τ) D s D r τl(, τ) D β r τ l (, τ) D s D r τb(, τ) Let L(, A ), B(, A ) and C(, A ), defined as in (3.1), be the oerators corresonding to the symbols l(, τ), b(, τ) and c(, τ), resectively, in Λ. Then we have L(, A ) = L(, A ) + L (, A ), B(, A ) = B(, A ) + B (, A ), C(, A ) = L(, A )B(, A ) = L(, A )B(, A ) + L (, A )B(, A ) + L(, A )B (, A ) + L (, A )B (, A ). (5.1) Furthermore, we denote l(, τ)b(, τ) = c(, τ), l (, τ)b (, τ) = f(, τ), l(, τ)b (, τ) = f 1 (, τ), and b(, τ)l (, τ) = f 2 (, τ). Then C(, A ) = C(, A ) + F(, A ) + F 1 (, A ) + F 2 (, A ). (5.2) Lemma 5.2. We have C(, A )φ = L(, A )B(, A )φ for φ L 2 (R + ; d ). Proof. From (3.1), (3.2) and C(, A ) = C(, A ) + C (, A ), we get as desired. (KC(, A )φ)(τ) = c(, τ)(kφ)(τ) = l(, τ)b(, τ)(kφ)(τ) = l(, τ)(kb(, A )φ)(τ) = (KL(, A )B(, A )φ)(τ), Lemma 5.3. We have F 1 (, A )φ = L(, A )B (, A )φ for φ L 2 (R + ; d ). Proof. Using (3.1), (3.2) and f 1 (, τ) =, we get (K(F 1 (, A )φ))(τ) = (K(F 1 (, A )φ))(τ).

10 4 A. Prasad and U. K. Mandal, Pseudo-differential oerators involving KL-transform Thus from (3.12), we have as desired. (KF 1 (, A )φ)(τ) = = K iτ ()f 1 (, τ)φ() d K iτ ()l(, τ)b (, τ)φ() d = l(, τ)(kb (, A )φ)(τ) = (KL(, A )B (, A )φ)(τ), Lemma 5.4. We have F 2 (, A )φ = B(, A )L (, A )φ for φ L 2 (R + ; d ). Proof. The roof is similar to the one of Lemma 5.3. Now, using (5.1) and (5.2), we get where L(, A )B(, A ) C(, A ) = L(, A )B(, A ) + L (, A )B(, A ) + L(, A )B (, A ) + L (, A )B (, A ) L(, A )B(, A ) L(, A )B (, A ) B(, A )L (, A ) F(, A ) = [L (, A ), B(, A )] + L (, A )B (, A ) F(, A ), [L (, A ), B(, A )] = L (, A )B(, A ) B(, A )L (, A ). Here [, ] reresents the commutators between the two oerators L (, A ) and B(, A ). The term F(, A ) reresents the.d.o. associated with the symbol f(, τ). Theorem 5.5. For φ L 2 (R + ; d ) we have where C n Proof. We have Using (3.3), we have [L (, A ), B(, A )]φ L 2 (R + ; d ) C n φ L 2 (R + ; d ), is some ositive constant and [, ] denotes the commutator between the two oerators. (KL (, A )B(, A )φ)(τ) = l τ(η)(kb(, A )φ)(η)η sinh(η) dη = l τ(η)b(, η)(kφ)(η)η sinh(η) dη. (K(L (, A )B(, A )φ))(τ) = (K(l (, τ)(k 1 b(, η)(kφ)(η))()))(τ), Thus using (3.5) and then (1.4), we get L (, A )B(, A )φ() = l (, τ)(k 1 (b(, η)(kφ)(η)))(). L (, A )B(, A )φ() C n (K 1 (b(, η)(kφ)(η)))(), L (, A )B(, A )φ L 2 (R + ; d ) C n K 1 (b(, η)(kφ)(η)) L 2 (R + ; d ) = C n b(, η)(kφ)(η) L 2 (R + ;η sinh(πη)) C n b(, 1) Kφ L 2 (R + ;η sinh(πη)) C n φ L 2 (R + ; d ).

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